Exponential Convergence through Linear Finite Element Discretization of Stratified Subdomains

TL;DR

This paper introduces the complex-length finite element method (CFEM), a novel discretization technique that achieves exponential convergence at specific points in a domain, significantly reducing computational costs for solving PDEs.

Contribution

The paper presents a new finite element discretization method that uses complex mesh bending and midpoint integration to achieve exponential convergence at targeted points.

Findings

Exponential convergence at pre-selected points in the domain.

Significant reduction in computational cost compared to traditional methods.

Effective for Laplace, Helmholtz, and elastodynamic equations.

Abstract

Motivated by problems where the response is needed at select localized regions in a large computational domain, we devise a novel finite element discretization that results in exponential convergence at pre-selected points. The two key features of the discretization are (a) use of midpoint integration to evaluate the contribution matrices, and (b) an unconventional bending of the mesh into complex space. Named complex-length finite element method (CFEM), the technique is linked to Pade approximants that provide exponential convergence of the Dirichlet-to-Neumann maps and thus the solution at specified points in the domain. Exponential convergence facilitates drastic reduction in the number of elements. This, combined with sparse computation associated with linear finite elements, results in significant reduction in the computational cost. The paper presents the basic ideas of the method as well as illustration of its effectiveness for a variety of problems involving Laplace, Helmholtz and elastodynamic equations.